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negative translation

It is well-known that classical propositional logic PLcc{}_{c} can be considered as a subsystem of intuitionistic propositional logic PLii{}_{i} by translating any wffAAA in PLcc{}_{c} into ¬⁢¬⁢AA\neg\neg A in PLii{}_{i}. According to Glivenko’s theorem, AAA is a theorem of PLcc{}_{c}iff¬⁢¬⁢AA\neg\neg A is a theorem of PLii{}_{i}. This translation, however, fails to preserve theoremhood in the corresponding predicatelogics. For example, if AAA is of the form ∃x⁢BxB\exists xB, then ⊢cAfragmentssubscriptprovescA\vdash_{c}A no longer implies⊢i¬¬AfragmentssubscriptprovesiA\vdash_{i}\neg\neg A. A number of translations have been devised to overcome this defect. They are collective known as negative translations or double negative translations of classical logic into intuitionistic logic. Below is a list of the most commonly mentioned negative translations: